IMPROVED RESULTS ON THE OSCILLATION OF THE MODULUS OF THE RUDIN-SHAPIRO POLYNOMIALS ON THE UNIT CIRCLE. September 12, 2018

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1 IMPROVED RESULTS ON THE OSCILLATION OF THE MODULUS OF THE RUDIN-SHAPIRO POLYNOMIALS ON THE UNIT CIRCLE Tamás Erdélyi September 12, 2018 Dedicated to Paul Nevai on the occasion of his 70th birthday Abstract. Let either R k (t := P k (e it 2 or R k (t := Q k (e it 2, where P k and Q k are the usual Rudin-Shapiro polynomials of degree n 1 with n = 2 k. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin- Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant A > 0 such that the equation R k (t = (1 + ηn has at least An distinct zeros in [0,2π whenever η is real, η < 2 11, and n is sufficiently large. In this paper we show that the equation R k (t = (1+ηn has at least (1/2 η εn/2 distinct zeros in [0,2π for every η ( 1/2,1/2, ε > 0, and sufficiently large k k η,ε. 1. Introduction Let D := {z C : z < 1} denote the open unit disk of the complex plane. Let D := {z C : z = 1} denote the unit circle of the complex plane. The Mahler measure M 0 (f is defined for bounded measurable functions f on D by ( 1 2π M 0 (f := exp log f(e it dt. 2π 0 It is well known, see [HL-52], for instance, that M 0 (f = lim q 0+ M q(f, Key words and phrases. polynomials, restricted coefficients, oscillation of the modulus on the unit circle, Rudin-Shapiro polynomials, oscillation of the modulus on the unit circle, number of real zeros in the period Mathematics Subject Classifications. 11C08, 41A17, 26C10, 30C15 1 Typeset by AMS-TEX

2 where M q (f := ( 1 2π f(e it 1/q dt q, q > 0. 2π 0 It is also well known that for a function f continuous on D we have M (f := max t [0,2π] f(eit = lim q M q(f. It is a simple consequence of the Jensen formula that for every polynomial of the form M 0 (f = c f(z = c n max{1, z j } j=1 n (z z j, c,z j C. j=1 See [BE-95, p. 271] or [B-02, p. 3], for instance. Let P c n be the set of all algebraic polynomials of degree at most n with complex coefficients. Let T n be the set of all real (that is, real-valued on the real line trigonometric polynomials of degree at most n. Finding polynomials with suitably restricted coefficients and maximal Mahler measure has interested many authors. The classes of Littlewood polynomials and the classes n L n := f : f(z = a j z j, a j { 1,1} j=0 n K n := f : f(z = a j z j, a j C, a j = 1 j=0 of unimodular polynomials are two of the most important classes considered. Observe that L n K n and M 0 (f M 2 (f = n+1, f K n. Beller and Newman [BN-73] constructed unimodular polynomials f n K n such that M 0 (f n n c/logn with an absolute constant c > 0. Section 4 of [B-02] is devoted to the study of Rudin-Shapiro polynomials. Littlewood asked if there were polynomials f nk L nk satisfying c 1 nk +1 f nk (z c 2 nk +1, 2 z D,

3 with some absolute constants c 1 > 0 and c 2 > 0, see [B-02, p. 27] for a reference to this problem of Littlewood. To satisfy just the lower bound, by itself, seems very hard, and no such sequence (f nk of Littlewood polynomials f nk L nk is known. A sequence of Littlewood polynomials that satisfies just the upper bound is given by the Rudin-Shapiro polynomials. The Rudin-Shapiro polynomials appear in Harold Shapiro s 1951 thesis[s-51] at MIT and are sometimes called just Shapiro polynomials. They also arise independently in Golay s paper [G-51]. They are remarkably simple to construct and are a rich source of counterexamples to possible conjectures. The Rudin-Shapiro polynomials are defined recursively as follows: P 0 (z := 1, Q 0 (z := 1, P k+1 (z := P k (z+z 2k Q k (z, Q k+1 (z := P k (z z 2k Q k (z, for k = 0,1,2,.... Note that both P k and Q k are polynomials of degree n 1 with n := 2 k having each of their coefficients in { 1,1}. In signal processing, the Rudin- Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. It is well known and easy to check by using the parallelogram law that Hence P k+1 (z 2 + Q k+1 (z 2 = 2( P k (z 2 + Q k (z 2, z D. (1.1 P k (z 2 + Q k (z 2 = 2 k+1 = 2n, z D. It is also well known (see Section 4 of [B-02], for instance, that and hence Q k ( z = P k (z = zn 1 P k (1/z, (1.2 Q k ( z = P k (z, z D. z D, Let K := R (mod 2π. Let m(a denote the one-dimensional Lebesgue measure of A K. In 1980 Saffari conjectured the following. Conjecture 1.1. Let P k and Q k be the Rudin-Shapiro polynomials of degree n 1 with n := 2 k. We have M q (P k = M q (Q k 2k+1/2 (q/2+1 1/q for all real exponents q > 0. Equivalently, we have } lim m t K : P k (e it 2 k 2 k+1 [α,β] } = lim m t K : Q k (e it 2 k 2 k+1 [α,β] = 2π(β α 3

4 whenever 0 α < β 1. This conjecture was proved for all even values of q 52 by Doche [D-05] and Doche and Habsieger [DH-04]. Recently B. Rodgers [R-16] proved Saffari s Conjecture 1.1 for all q > 0. See also [EZ-17]. An extension of Saffari s conjecture is Montgomery s conjecture below. Conjecture 1.2. Let P k and Q k be the Rudin-Shapiro polynomials of degree n 1 with n := 2 k. We have lim m t K : P k(e it } E k 2 k+1 = lim m t K : Q k(e it } E = 2m(E k 2 k+1 for any measurable set E D := {z C : z < 1}. B. Rodgers [R-16] proved Montgomery s Conjecture 1.2 as well. 2. New Results Let either R k (t := P k (e it 2 or R k (t := Q k (e it 2. In [AC-18] we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin- Shapiro polynomials with a deep theorem of Littlewood (see Theorem 1 in [L-66] to prove that there is an absolute constant A > 0 such that the equation R k (t = (1 + ηn with n := 2 k has at least An distinct zeros in K whenever η is real, η 2 11, and n is sufficiently large. In this paper we improve this result substantially. Theorem 2.1. The equation R k (t = n has at least n/4+1 distinct zeros in K. Moreover, with the notation t j := 2πj/n, there are at least n/2+2 values of j {0,1...,n 1} for which the interval [t j,t j+1 ] has at least one zero of the equation R k (t = n. Theorem 2.2. The equation R k (t = (1+ηn has at least (1/2 η εn/2 distinct zeros in K for every η ( 1/2,1/2, ε > 0, and sufficiently large k k η,ε. 3. Lemma In the proof of Theorem 2.1 we need the lemma below stated and proved as Lemma 3.1 in [E-16]. Lemma 3.1. Let n 2 be an integer, n := 2 k, and let We have z j := e it j, t j := 2πj n, j Z. P k (z j = 2P k 2 (z j, j = 2u, u Z, P k (z j = ( 1 (j 1/2 2iQ k 2 (z j, j = 2u+1, u Z, where i is the imaginary unit. 4

5 4. Proofs Proof of Theorem 2.1. Let k 2 be an integer. Assume that R k (t = P k (e it 2, the proof in the case R k (t = Q k (e it 2 is the same. For the sake of brevity let A j := R k 2 (t j n/4, j = 0,1,...,n. Using the notation of Lemma 3.1 we study the (n + 1-tuple A 0,A 1,...,A n. Observe that R k 2 is a real trigonometric polynomial of degree n/4 1 = 2 k /4 1, and hence R k 2 (t n/4 has at most n/2 2 zeros in K. Therefore the Intermediate Value Theorem yields that the number of sign changes in the (n + 1-tuple A 0,A 1,...,A n is at most n/2 2. Thus there are integers with m n (n/2 2 = n/2+2 such that 0 j 1 < j 2 < < j m n 1 (4.1 A jν A jν +1 0, ν = 0,1,...,m. Using Lemma 3.1 we have either (4.2 16A jν A jν +1 =(4(R k 2 (t jν n/4(4(r k 2 (t jν +1 n/4 =(4 P k 2 (e it jν 2 n(4 P k 2 (e it jν+1 2 n =( P k (e it jν 2 n( Q k (e it jν+1 2 n =( P k (e it jν 2 n(n P k (e it jν+1 2, or (4.3 16A jν A jν +1 =(4(R k 2 (t jν n/4(4(r k 2 (t jν +1 n/4 =(4 P k 2 (e it jν 2 n(4 P k 2 (e it jν+1 2 n =( Q k (e it jν 2 n( P k (e it jν+1 2 n =( n P k (e it jν 2 ( P k (e it jν+1 2 n. Combining (4.1, (4.2, and (4.3, we can deduce that ( P k (e it jν 2 n( P k (e it jν+1 2 n = 16A jν A jν +1 0, ν = 0,1,...,m. Hence the the Intermediate Value Theorem implies that R k (t n = P k (e it 2 n has at least one zero in each of the intervals [t jν,t jν +1], ν = 0,1,...,m. Recalling that m n/2 + 2 we conclude that R k (t n = P k (e it 2 n has at least m/2 = n/4+1 distinct zeros in K. 5

6 Proof of Theorem 2.2. This follows from the proof of Theorem 2.1 combined with B. Rodgers s resolution of Saffari s Conjecture 1.1. Assume that R k (t = P k (e it 2, the proof in the case R k (t = Q k (e it 2 is the same. Also, we may assume that η > 0, the case η = 0 is contained in Theorem 2.1. We use the notation in the proof of Theorem 2.1. Recall that each of the intervals [t jν,t jν +1], ν = 0,1,...,m, has at least one zero of R k. On the other hand, by Saffari s Conjecture proved by Rodgers [R-16] we have mt K : R k (t n η n} < 2π(1+ε η for every η ( 1/2,1/2, ε > 0, and sufficiently large k k η,ε. Hence, with the notation B η := {t K : R k (t n η n}, there are at least m (1+ε η n distinct values of ν {1,2,...,m} such that [t jν,t jν +1]\B η for every η ( 1/2,1/2, and sufficiently large k k η,ε. Hence by the Intermediate Value Theorem there are at least m (1+ε η n distinct values of ν {1,2,...,m} for which R k (t n = η n has a zero in (t jν,t jν +1 for every η ( 1/2,1/2, ε > 0, and sufficiently large k k η,ε. Now observe that (1.1 and (1.2 imply that that is, P k (z 2 + P k ( z 2 = 2n, z D, R k (t n = n R k (t+π, t K. Hence for every η ( 1/2,1/2 the number of distinct zeros of R k (t n = η n in K is exactly twice the number of distinct zeros of R k (t = (1+ηn in K. We conclude that there are at least 1 (m (1+εηn (1/2 η εn/2 2 distinct values of ν {1,2,...,m} for which R k (t n = ηn has a zero in (t jν,t jν +1 for every η ( 1/2,1/2, ε > 0, and sufficiently large k k η,ε. 5. Acknowledgement The author thanks Stephen Choi for checking the details of this paper carefully before its submission. References AC-17. J.-P. Allouche, K.-K. S. Choi, A. Denise, T. Erdélyi, and B. Saffari, Bounds on autocorrelation coefficients of Rudin-Shapiro polynomials, manuscript. B-02. P. Borwein, Computational Excursions in Analysis and Number Theory, Springer, New York,

7 BE-95. P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, New York, BN-73. E. Beller and D.J. Newman,, An extremal problem for the geometric mean of polynomials, Proc. Amer. Math. Soc. 39 (1973, D-05. Ch. Doche, Even moments of generalized Rudin-Shapiro polynomials, Math. Comp. 74 (2005, no. 252, DH-04. Ch. Doche and L. Habsieger, Moments of the Rudin-Shapiro polynomials, J. Fourier Anal. Appl. 10 (2004, no. 5, E-16. T. Erdélyi, The Mahler measure of the Rudin-Shapiro polynomials, Constr. Approx. 43 (2016, no. 3, EZ-17. S.B. Ekhad and D. Zeilberger, Integrals involving Rudin-Shapiro polynomials and sketch of a proof of Saffari s conjecture, To appear in the Proceedings of the Alladi60 conference. G-51. M.J. Golay, Static multislit spectrometry and its application to the panoramic display of infrared spectra,, J. Opt. Soc. America 41 (1951, HL-52. G.H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, London, L-66. J.E. Littlewood, The real zeros and value distributions of real trigonometrical polynomials, J. London Math. Soc. 41 (1966, R-16. B. Rodgers, On the distribution of Rudin-Shapiro polynomials and lacunary walks on SU(2, to appear in Adv. Math., arxiv.org/abs/ S-51. H.S. Shapiro, Extremal problems for polynomials and power series, Master thesis, MIT, Department of Mathematics, Texas A&M University, College Station, Texas 77843, College Station, Texas address: terdelyi@math.tamu.edu 7